HAJEK-RENYI-TYPE INEQUALITY FOR SOME NONMONOTONIC FUNCTIONS OF ASSOCIATED RANDOM VARIABLES ISHA DEWAN AND B. L. S. PRAKASA RAO Received 21 April 2005; Revised 26 October 2005; Accepted 11 December 2005 Let {Y n , n ≥ 1} be a sequence of nonmonotonic functions of associated random variables. We derive a Newman and Wright (1981) type of inequality for the maximum of partial sums of the sequence {Y n , n ≥ 1} and a Hajek-Renyi-type inequality for nonmonotonic functions of associated random variables under some conditions. As an application, a strong law of large numbers is obtained for nonmonotonic functions of associated ran- dom varaibles. Copyright © 2006 I. Dewan and B. L. S. P. Rao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, dis- tribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Let {Ω,Ᏺ,ᏼ} aprobabilityspaceand{X n , n ≥ 1} beasequenceofassociatedrandom variables defined on it. A finite collection {X 1 ,X 2 , ,X n } is said to be associated if for every pair of functions h(x)andg(x)from R n to R, which are nondecreasing componen- twise, Cov h(X),g(X) ≥ 0, (1.1) whenever it is finite, where X = (X 1 ,X 2 , ,X n ). The infinite sequence {X n , n ≥ 1} is said to be associated if ever y finite subfamily is associated. Associated random variables are of considerable interest in reliability studies (cf. Bar- low and Proschan [1], Esary et al. [6]), statistical physics (cf. Newman [9, 10]), and perco- lation theory (cf. Cox and Grimmet [ 4]). For an extensive review of several probabilistic and statistical results for associated sequences, see Roussas [14] and Dewan and Rao [5]. Newman and Wright [12] proved an inequality for maximum of partial sums and Prakasa Rao [13] proved the Hajek-Renyi-type inequality for associated random vari- ables. Esary et al. [6] proved that monotonic functions of associated random variables are associated. Hence one can easily extend the above-mentioned inequalities to monotonic Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2006, Article ID 58317, Pages 1–8 DOI 10.1155/JIA/2006/58317 2 Hajek-Renyi type inequality functions of associated random variables. We now generalise the above results to some nonmonotonic functions of associated random variables. In Section 2, we discuss some preliminaries. Two inequalities are proved for non- monotonic functions of associated random variables in Section 3.Asanapplication,a strong law of large numbers is derived for nonmonotonic functions of associated ran- dom variables in Section 4. 2. Preliminaries Let us discuss some definitions and results which will be useful in proving our main results. Definit ion 2.1 (Newman [11]). Let f and f 1 be two real-valued functions defined on R n . Then f f 1 if and only if f 1 + f and f 1 − f are both nondecreasing componentwise. In particular, if f f 1 ,then f 1 will be nondecreasing componentwise. Dewan and Rao [5] observed the following. Remark 2.2. Suppose that f is a real-valued function defined on R.Then f f 1 for some real-valued function defined f 1 on R if and only if for x<y, f (y) − f (x) ≤ f 1 (y) − f 1 (x), f (x) − f (y) ≤ f 1 (y) − f 1 (x) . (2.1) It is clear that these relations hold if and only if, for x<y, f (y) − f (x) ≤ f 1 (y) − f 1 (x) . (2.2) Remark 2.3. If f is a Lipschitzian function defined on R, that is, there exists a positive constant C such that f (x) − f (y) ≤ C|x − y|, (2.3) then f f ,with f (x) = Cx. (2.4) In general, if f is a Lipschitzian function defined on R n ,then f f ,where f x 1 , ,x n = Lip( f ) n i=1 x i ,Lip(f ) = sup x=y f x 1 , ,x n − f y 1 , , y n n i =1 x i − y i < ∞. (2.5) Let {X n , n ≥ 1} be a sequence of associated random variables. Let (i) Y n = f n X 1 ,X 2 , , (ii) Y n = f n X 1 ,X 2 , , (iii) f n f n , (iv) E Y 2 n < ∞, E Y 2 n < ∞,forn ∈ N. (2.6) I. Dewan and B. L. S. P. Rao 3 For convenience, we write that Y n Y n if the conditions stated in (i)–(iv) hold. The functions f n , f n are assumed to be real-valued and depend only on a finite number of X n ’s. Let S n = n k =1 Y k , S n = n k =1 Y k . Matula [8] proved the following result which will be useful in proving our results. He used them to prove the strong law of large numbers and the central limit theorem for nonmonotonic functions of associated random variables. Lemma 2.4. Suppose the conditions stated above in (2.6)hold.Then (i) Var f n ≤ Var f n , (ii) Cov f n , f n ≤ Var f n , (iii) Var S n ≤ Var S n , (iv) f 1 + f 2 + ···+ f n f 1 + f 2 + ···+ f n , (v) Cov f 1 + f 1 , f 2 + f 2 ≤ 4Cov f 1 , f 2 , (vi) Cov f 1 − f 1 , f 2 − f 2 ≤ 4Cov f 1 , f 2 . (2.7) For completeness, now state the inequalities due to Newman and Wright [12] and Prakasa Rao [13] for associated random variables. Lemma 2.5 (Newman and Wright). Suppose X 1 ,X 2 , ,X m are associated, mean zero, finite variance random variables, and M ∗ m = max(S ∗ 1 ,S ∗ 2 , ,S ∗ m ),whereS ∗ n = n i =1 X i . Then E M ∗ m 2 ≤ Var S ∗ m . (2.8) Remark 2.6. Note that if X 1 ,X 2 , ,X m are associated random variables, then −X 1 ,−X 2 , , −X m also form a set of associated random variables. Let M ∗∗ m = max(−S ∗ 1 ,−S ∗ 2 , , −S ∗ m )and M ∗ m = max(|S ∗ 1 |,|S ∗ 2 |, ,|S ∗ m |). Then M ∗ m = max(M ∗ m ,M ∗∗ m )and( M ∗ m ) 2 ≤(M ∗ m ) 2 +(M ∗∗ m ) 2 so that E M ∗ m 2 ≤ 2Var S ∗ m . (2.9) Lemma 2.7 (Prakasa Rao). Let {X n , n ≥ 1} be an associated sequence of random variables with Var(X n ) = σ 2 n < ∞, n ≥ 1,and{b n , n ≥ 1} a positive nondecreasing sequence of real numbers. Then, for any > 0, P max 1≤k≤n 1 b n k i=1 X i − E X i ≥ ≤ 4 2 ⎡ ⎣ n j=1 Var X j b 2 j + 1≤ j=k≤n Cov X j ,X k b j b k ⎤ ⎦ . (2.10) 3. Main results We now extend the Newman and Wright’s [12] result to nonmonotonic functions of as- sociated random variables satisfying conditions (2.6). 4 Hajek-Renyi type inequality Theorem 3.1. Let Y 1 ,Y 2 , ,Y m be as defined in (2.6) with zero-mean and finite variances. Let M m = max(|S 1 |,|S 2 |, ,|S m |). Then E M 2 m ≤ (20)Var S m . (3.1) Proof. Observe that max 1≤k≤m |S k |= max 1≤k≤m S k − S k − E S k − S k + E S k ≤ max 1≤k≤m S k − S k − E S k +max 1≤k≤m S k − E S k . (3.2) Note that S k − E( S k )and S k − S k − E( S k ) are partial sums of associated random variables each with mean zero. Hence using the results of Newman and Wright [12], we get that E M 2 m ≤ E max 1≤k≤m S k 2 ≤ 2 E max 1≤k≤m S k − S k − E S k 2 + E max 1≤k≤m S k − E S k 2 ≤ 4 Var S m − S m +Var S m (by Remark 2.6) ≤ 4 Var 2 S m +Var S m = 20Var S m . (3.3) We have used the fact that Var 2 S n = Var S n − S n + S n + S n = Var S n − S n +Var S n + S n +2Cov S n + S n , S n − S n . (3.4) Since S n + S n and S n − S n are nondecreasing functions of associated random variables, it follows that Cov( S n + S n , S n − S n ) ≥ 0. Hence Var(2 S n ) ≥ Var( S n − S n ). We now prove a Hajek-Renyi-type inequality for some nonmonotonic functions of associated random variables satisfying conditions (2.6). Theorem 3.2. Let {Y n , n ≥ 1} be sequence of nonmonotonic functions of assoc iated random variables as defined in (2.6). Suppose that Y n Y n , n ≥ 1.Let{b n , n ≥ 1} beapositive nondecreasing sequence of real numbers. Then for any > 0, P max 1≤k≤n 1 b n k i=1 Y i − E Y i ≥ ≤ (80) −2 ⎡ ⎣ n j=1 Var Y j b 2 j + 1≤ j=k≤n Cov Y j , Y k b j b k ⎤ ⎦ . (3.5) I. Dewan and B. L. S. P. Rao 5 Proof. Let T n = n j =1 (Y j − E(Y j )). Note that P max 1≤k≤n T k b k ≥ = P max 1≤k≤n T k − T k − E T k − T k + E T k b k ≥ ≤ P max 1≤k≤n T k − T k − E T k b k ≥ 2 + P max 1≤k≤n T k − E( T k ) b k ≥ 2 ≤ (16) −2 n j=1 Var Y j − Y j b 2 j + 1≤ j=k≤n Cov Y j − Y j , Y k − Y k b j b k + (16) −2 n j=1 Var Y j b 2 j + 1≤ j=k≤n Cov Y j , Y k b j b k . (3.6) The result follows by applying the following inequalities: Var Y j − Y j ≤ 4Var Y j ,Cov Y j − Y j , Y k − Y k ≤ 4Cov Y j , Y k . (3.7) 4. Applications Let C denote a generic positive constant. Corollary 4.1. Let {Y n , n ≥ 1} be sequence of nonmonotonic functions of associated ran- dom variables satisfying the conditions in (2.6). Assume that ∞ j=1 Var Y j + 1≤ j=k<∞ Cov Y j , Y k < ∞. (4.1) Then ∞ j=1 (Y j − EY j ) converges almost surely. Proof. Without loss of generality, assume that EY j = 0forall j ≥ 1. Let T n = n j =1 Y j and > 0. Using Theorem 3.2 is easy to see that P sup k,m≥n T k − T m ≥ ≤ P sup k≥n T k − T n ≥ 2 + P sup m≥n T m − T n ≥ 2 ≤ C limsup N→∞ P sup n≤k≤N T k − T n ≥ 2 ≤ C −2 ∞ j=n Var Y j + n≤ j=k<∞ Cov Y j , Y k . (4.2) 6 Hajek-Renyi type inequality The last term tends to zero as n →∞because of (4.1). Hence the sequence of random variables {T n , n ≥ 1} is Cauchy almost surely which implies that T n converges almost surely. The following corollary proves the strong law of large numbers for nonmonotonic functions of associated random variables. Corollary 4.2. Let {Y n , n ≥ 1} be sequence of nonmonotonic functions of associated ran- dom variables satisfying the conditions in (2.6). Suppose that ∞ j=1 Var Y j b 2 j + 1≤ j=k<∞ Cov Y j , Y k b j b k < ∞. (4.3) Then (1/b n ) n j =1 (Y j − EY j ) converges to zero almost surely as n →∞. Proof. The proof is an immediate consequence of Theorem 3.2 and the Kronecker lemma (Chung [3]). Remark 4.3. Birkel [2] proved a strong law of large numbers for positively dependent ran- dom variables. Prakasa Rao [13] proved a strong law of large numbers for associated se- quences as a consequence of the Hajek-Renyi-type inequality. Marcinkiewicz-Zygmund- type strong law of large numbers for associated random variables, for which the second moment is not necessarily finite, was studied in Louhichi [7]. Strong law of large numbers for monotone functions of associated sequences follows from these results since mono- tone functions of associated sequences are associated. However Corollary 4.2 gives suffi- cient conditions for the strong law of large numbers to hold for possibly nonmonotonic functions of associated sequences whose second moments are finite. For any random variable X and any constant k>0, define X k = X if |X|≤k, X k =−k if X< −k,andX k = k if X>k.The following theorem is an analogue of the three series theorem for nonmonotonic functions of associated random variables. Corollary 4.4. Let {Y n , n ≥ 1} be sequence of nonmonotonic functions of associated ran- dom variables. Further suppose that there exists a constant k>0 such that Y k n Y k n satisfy- ing the conditions in (2.6)and ∞ n=1 P Y n ≥ k < ∞, ∞ n=1 E Y k n < ∞, ∞ j=1 Var Y k j + 1≤ j=j <∞ Cov Y k j , Y k j < ∞. (4.4) Then ∞ n=1 Y n converges almost surely. I. Dewan and B. L. S. P. Rao 7 Corollary 4.5. Let {Y n , n ≥ 1} be sequence of nonmonotonic functions of associated ran- dom variables satisfying the conditions in (2.6). Suppose ∞ j=1 Var Y j b 2 j + 1≤ j=k<∞ Cov Y j , Y k b j b k < ∞. (4.5) Let T n = n j =1 (Y j − E(Y j )).Then,forany0 <r<2, E sup n T n b n r < ∞. (4.6) Proof. Note that E sup n T n b n r < ∞ (4.7) if and only if ∞ 1 P sup n T n b n r >t 1/r dt < ∞. (4.8) The last inequality holds because of Theorem 3.2 and condition (4.5). Hence the result stated in (4.6)holds. Acknowledgment The authors thank the referee for the comments and suggestions which have led to an improved presentation. References [1] R. E. Barlow and F. Proschan, Statistical Theory of Reliability and Life Testing: Probability Models, Holt, Rinehart and Winston, New York, 1981. [2] T. Birkel, A note on the strong law of large numbers for positively dependent random variables, Statistics & Probability Letters 7 (1988), no. 1, 17–20. [3] K.L.Chung,A Course in Probability Theory, Academic Press, New York, 1974. [4] J. T. Cox and G. Grimmett, Central limit theorems for associated random variables and the perco- lation model, The Annals of Probability 12 (1984), no. 2, 514–528. [5] I. Dewan and B. L. S. Prakasa Rao, Asymptot ic normality for U-statistics of associated random variables, Journal of Statistical Planning and Inference 97 (2001), no. 2, 201–225. [6] J.Esary,F.Proschan,andD.Walkup,Association of random variables, with applications, Annals of Mathematical Statistics 38 (1967), 1466–1474. [7] S. Louhichi, Convergence rates in the strong law for associated random variables, Probability and Mathematical Statistics 20 (2000), no. 1, 203–214. [8] P. Matula, Limit theorems for sums of nonmonotonic functions of associated random variables, Journal of Mathematical Sciences 105 (2001), no. 6, 2590–2593. [9] C.M.Newman,Normal fluctuations and the FKG inequalities, Communications in Mathematical Physics 74 (1980), no. 2, 119–128. [10] , A general central limit theorem for FKG systems, Communications in Mathematical Physics 91 (1983), no. 1, 75–80. 8 Hajek-Renyi type inequality [11] , Asymptotic independence and limit theorems for positively and negatively dependent ran- dom variables, Inequalities in Statistics and Probability (Lincoln, Neb, 1982) (Y. L. Tong, ed.), vol. 5, Institute of Mathematical Statistics, California, 1984, pp. 127–140. [12] C. M. Newman and A. L. Wright, An invariance principle for certain dependent sequences,The Annals of Probability 9 (1981), no. 4, 671–675. [13] B. L. S. Prakasa Rao, Hajek-Renyi-type inequality for associated sequences, Statistics & Probability Letters 57 (2002), no. 2, 139–143. [14] G. G. Roussas, Positive and negative dependence with some statistical applications,Asymptotics, Nonparametrics, and Time Series (S. Ghosh, ed.), vol. 158, Marcel Dekker, New York, 1999, pp. 757–788. Isha Dewan: Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, New Delhi 110016, India E-mail address: isha@isid2.isid.ac.in B. L. S. Prakasa Rao: Department of Mathematics and Statistics, University of Hyderabad, Hyderabad 500046, India E-mail address: blsprsm@uohyd.ernet.in . Hajek-Renyi-type inequality for nonmonotonic functions of associated random variables under some conditions. As an application, a strong law of large numbers is obtained for nonmonotonic functions of associated. Hajek-Renyi type inequality functions of associated random variables. We now generalise the above results to some nonmonotonic functions of associated random variables. In Section 2, we discuss some preliminaries proves the strong law of large numbers for nonmonotonic functions of associated random variables. Corollary 4.2. Let {Y n , n ≥ 1} be sequence of nonmonotonic functions of associated ran- dom variables